An Investigation of Tip-Vortex Turbulence Structure using Large-EddySimulation

1David Hanson ∗; 1Michael Kinzel; 1Kyle Sinding; and 1Michael Krane

1The Pennsylvania State University, University Park, PA, USAAbstractIn this effort, the tip vortex behind a rectangular plan-form lifting fin with a cavitation-relevant airfoilsection (NACA-66 mod) is studied using Large Eddy Simulation (LES) at a free-stream Reynoldsnumber of 500,000. Elementary flow quantities (lift, drag, velocities) are compared with measurementsmade in the Garfield Thomas Water Tunnel (GTWT) 12-inch circular test-section. Good agreement isseen with the measured lift and drag. The characteristic 45-degree lag between the mean strain rateand the Reynolds stress components in the plane of the tip vortex is observed. The calculated flow fieldprovides insight into the turbulent structure of the tip vortex for this fin.Keywords: tip-vortex, turbulence, computational fluid dynamics

Introduction

Cavitation within tip-vortices, i.e., Tip-Vortex Cavitation (TVC), is important aspect relevant to the design ofpropellers for underwater vehicles [1]. One specific adverse aspect is erosion associated when a cavitating vorteximpacts downstream surfaces. Understanding the complicated turbulent flow-field in the region of TVC inceptionis necessary in order to develop Computational Fluid Dynamics (CFD) models to predict TVC.

In the present literature, the authors found a limited database of detailed turbulent structure information withintip vortices. The turbulence, however, directly informs the overall tip-vortex evolution and the development ofTVC. The cavitation behavior within a tip vortex is informed by the turbulent pressure and velocity fluctuations.The challenges involved in measuring experimentally both the time-varying pressure and velocity fields withina tip vortex are considerable. As such, the established understanding of the correlation between these quantitiesis limited. The present work involves a numerical study using Large-Eddy Simulation (LES) of a tip-vortexflow around a rectangular fin to gain detailed insight into the resulting turbulent flow-field. The tip-vortex flowgeometry in question was recently studied experimentally by Sinding and Krane [2].

For the present work, the authors will interrogate the unsteady LES flow-field solution to extract the fine detailsof the turbulent structure. LES will, thus, be used to develop a thorough understanding of Reynolds-AveragedNavier-Stokes (RANS) turbulence model failure in tip-vortex flows.

Description of the Flow

The fin geometry considered in this work was studied experimentally by Sinding and Krane [2]. The chord is 4in(0.1016m) and the aspect ratio is two. The fin has an NACA-66(mod) thickness distribution [3] wrapped aroundan NACA 4-digit camber line. The maximum camber in the chord-wise direction is 0.0192 × chord, located at50% chord. The camber is uniform along the span and was chosen such that at zero incidence the lift coefficient is0.4 with a span-wise lift distribution that is approximately elliptical. A similar fin was also studied experimentallyby Hanson [4]; however, the fin at that time had a geometry with a rounded tip. Sinding and Krane modified thefin model to have a flat tip in order to simplify the boundary-layer detachment as the flow enters the tip vortex.Because the shape of the fin tip is expected to influence strongly the tip-vortex dynamics, this paper focuses onlyon the flat-tipped fin studied by Sinding and Krane. The physical fin model is shown in the 12-inch (0.3048m)circular test-section water tunnel at Penn State’s Applied Research Laboratory in Figure 1.

The present computational model includes the fin geometry and the circular water tunnel section. The free-streamspeed is 15 fts (4.57ms ), yielding a moderate Reynolds number of∼ 5×105. The tunnel is modeled in order for thecalculated lift and drag to be directly comparable with the experimental measurements without correcting for the

∗Corresponding Author, David Hanson: drh5013@psu.edu

1

Figure 1: Side view of the fin in the water tunnel. TVC is visible near the top of the figure downstream of the tip.Flow is right-to-left.

blockage or the presence of walls. The present work uses a geometry and computational mesh as correspondingto that presented in Figure 2.

Methods

Flow Solver

In the present work, the discretized LES equations of fluid motion are solved in the context of the commercialCFD suite Star-CCM+ [5]. Star-CCM+ is developed and maintained by CD-Adapco, a division of Siemens Coor-poration. Star-CCM+ is capable of many different spatial and temporal solution methods; only those relevant tothe present work are discussed here.

The spatial- and temporal-discretization schemes used are formally second-order accurate. Pressure-velocity cou-pling in the present incompressible flow is accomplished at each time step in a segregated manner via the SIMPLEalgorithm, wherein a discrete pressure-Poisson equation provides an update to the solution of the discretized mo-mentum conservation equations within each linear sub-iteration. All solution variables are located at the cellcenters; facial fluxes are approximated via a Total-Variation Bounding (TVB) central-difference scheme.

Equations of Fluid Motion

The subgrid-filtered equations of mass- and momentum-conservation for LES of an incompressible flow are writ-ten as

∂uj∂xj

= 0 (1)

and∂ui∂t

+∂

∂xj

(uiuj + δijp− T ij

)=∂τ rij∂xj

(2)

where the overline in this context refers to variables that have undergone the subgrid filter.

Formally, the residual stress tensor τ rij = uiuj − uiuj represents contributions from three types of interactions:interactions among resolved scales, interactions among the unresolved scales, and cross-interactions betweenresolved and unresolved scales. These are typically called the Leonard stress, the Reynolds stress, and theClark stress, respectively. In practice, however, this term is modeled via a Boussinesq approximation, τ rij =

2

Figure 2: Top: geometry of the flow over the finite-span fin in the 12in (0.3048m) circular test section of thewater-tunnel. The z− direction points into the page and completes the right-handed set. The origin is coincident

with where the fin leading edge meets the tunnel wall. Bottom: the refinement regions overlayed upon the fingeometry.

2 (νt,SGSSij − δijkSGS/3), where kSGS is the subgrid-scale turbulent kinetic energy. The subgrid turbulent vis-cosity, νt,SGS , is given by a simple function evaluation in terms of the local strain rate and the mesh size. TheWall-Adapting Large Eddy (WALE) constitutive model is used to define νt,SGS in the present work [6].

Discretization

The flow equations of motion are discretized and solved on an unstructured trimmed Cartesian mesh with prismlayers. This mesh type was chosen because it allows the best compromise between the competing goals of mini-mizing the numerical dissipation, minimizing the total cell count required to resolve the geometry, and minimizingthe mesh generation time. Within the high-resolution turbulence-resolving region, the mesh is uniform and rec-tilinear with the exception of the thin near-wall prism layer. Such a uniform mesh reduces the importance ofnumerical diffusion and errors due to the gradient reconstruction.

The computational mesh used in the present study has approximately 70× 106 cells. As mentioned by Georgiadiset al. [7], the spatial resolution of the mesh used for LES is typically determined by the maximum limit of thecomputing resources available. That is also the case here. Rather than resolving the viscous sublayer, the lowestregion of the boundary layer at the surface of the fin is approximated using a blended wall function based onReichardt’s law. The dimensionless distance, y+ = ∆yu∗/ν, from the wall to the nearest cell center, is between20 and 100 on the fin surface. In outer variables, this yields a wall-normal cell size of ∆y/δ ≈ 0.02, consistentwith the recommendation of Larsson et al. [8] for LES. Because this is a relatively simple wing geometry, withoutstrong pressure gradients or separated flow regions, use of wall functions is not expected to be detrimental to thesolution quality. The linear size of wall cells in the wall-parallel direction is 0.25 mm, or x+,z+ ∼ 50. The near-wall region of mesh refinement is shown in Figure 2. The present mesh provides approximately 40 grid pointsacross the vortex core diameter within the region of highest refinement. Using the computed solution, the Taylormicroscale (λ) is approximated and compared to the computational cell size in the Results section.

The time step is chosen such that the convective Courant number is less than unity in almost all computationalcells. For the present flow, a time step of 25 microseconds satisfies this requirement. Ten linear iterations areperformed within each non-linear time step. The solution was run for a total simulation time of 0.55 seconds;this corresponds to a simulation time of approximately 24 convective time scales. Time-averages for the various

3

P −u′ju′k∂uj

∂xi− u′iu′j

∂uk

∂xi

DS −2ν∂u′

k

∂xi

∂u′j

∂xi

PSC p′

ρ

(∂u′

j

∂xk+

∂u′k

∂xj

)DF1

∂∂xi

(−u′iu′ju′k

)DF2 ν ∂2

∂xi∂xi

(u′ju′k

)DF3

∂∂xi

(−p′ρ

(δiju′k + δiku′j

))

PTKE −u′iu′j∂uj

∂xi

ε νs′ijs′ij

DFTKE,1,pressure∂∂xi

(−u′i

pρ′)

DFTKE,1∂∂xi

(−u′iu′ju′j

)DFTKE,2 ν ∂2k

∂xi∂xi

DFTKE,3 ν ∂2

∂xi∂xi

(u′ju′j

)Table 1: Meaning of terms in RS equation (left) and the TKE equation (right)

quantities discussed in the Results section were recorded for approximately the last quarter second, or half thetotal simulation time, in order to avoid contaminating the averages with flow transients.

Turbulence Quantities

A goal of this study is to provide quantitative information regarding the detailed turbulent structure within thetip vortex as it undergoes the wrap-up process. Toward this end, every term both in the Reynolds Stress (RS)equation and in the much simpler Turbulent Kinetic Energy (TKE) equation is recorded at every fourth time stepon two planes (at one-quarter and one chord downstream of the trailing edge) across the tip vortex. Furthermore,time-averaged values of every term in the TKE equation are recorded at every cell in the domain.

In the context of a Cartesian coordinate system, the transport equation for each component of the RS tensor for anincompressible Newtonian flow is:

∂

∂t

(u′ju′k

)+ ui

∂

∂xi

(u′ju′k

)= PRS +DSRS + PSCRS +DFRS,1 +DFRS,2 +DFRS,3 (3)

where the symbols on the right side of the equation represent tensors that express turbulence production, pseudo-dissipation, the correlation between pressure and strain, and three types of diffusion, respectively. The forms forthese terms used in the present work are summarized in Table 1. Each term in this table is a symmetric secondorder tensor and therefore has six independent components that must be recorded.

A common simplification of the RS tensor equation is to assume that the dynamics of turbulence is controlledby the trace of the RS tensor, interpreted as the TKE, k = 1

2

(u′2 + v′2 + w′2

). This reduces the six tensor RS

equations to a scalar equation for k:

∂k

∂t+ ui

∂k

∂xi= PTKE − ε+DFTKE,1,pressure +DFTKE,1 +DFTKE,2 +DFTKE,3 (4)

where the right-hand-side symbols represent turbulence production, dissipation, and four types of diffusion, re-spectively. These are also summarized in Table 1.

The approach taken for the present work is to track the instantaneous and mean values of the five field variablesfor the TKE equation and the instantaneous values for the further 36 variables for the RS equation. This is inaddition to seven field variables for RS and TKE. All instantaneous variables in the RS equation were exportedevery fourth time step on two planes normal to the tip vortex located 0.25× chord and 1× chord downstream ofthe trailing edge, respectively; time-averaged quantities for the RS equation are therefore available only on theseplanes.

Results

Validation

The lift- and drag-coefficients, i.e., CL and CD, respectively, computed for the present flow are compared withexperimental measurements made by Hanson for this fin [4] in Figure 3. Recall that the fin used in those experi-

4

Figure 3: Top Row: CL from the present study compared with the measurements of Hanson [4]. CDmeasurements from that study are compared with the present CFD in the rightmost figure. Bottom Row: spectraof turbulent velocity fluctuations at two locations in the flow: immediately inboard (left) and outboard (right) ofthe vortex core; both probes are located one inch downstream of the trailing edge. The specific kinetic energy in

the flow, E = 12u

2, is made dimensionless by the local velocity variance, (u′2)rms.

mental measurements had a rounded, rather than CFD model having a flat-tip geometry. It is not anticipated thatthe tip geometry greatly to alterCL andCD. Agreement between the present CFD and experimental measurementsis quite close.

TKE spectra are shown for two different point probes within the tip vortex in Figure 3. The velocity point probesare located on the inboard and outboard edges of the tip vortex, respectively, 1in (0.0254m) downstream of thetrailing edge. The expected f−5/3 decay of the kinetic energy with respect to frequency is observed at higherfrequencies at both locations.

In the present work, λ is approximated from the flow solution as λ ∼√

10νk/ε. The ratio of this length scaleto the computational cell linear size, h, is shown in Figure 4 on a planar slice through the tip vortex at an axiallocation one chord downstream of the trailing edge. It is seen that the present mesh has approximately five cellsper λ within the wake and at the edge of the vortex core.

Elementary Turbulence Quantities

For the sake of brevity, only one elementary turbulence quantity is mentioned here. The rightmost image in Figure4 compares the y − z component of the mean strain rate tensor, Sij , with the corresponding component of thespecific Reynolds stress tensor τij . Chow [9] observed that the Reynolds shear stress in the plane of the tip vortex

5

Figure 4: Left: ratio of the cell linear size, h, to the approximate λ. Right: comparison of the mean shear ratecomponent S23 with the corresponding Reynolds stress component τ23.

tends to lag the corresponding component of the mean strain rate by 45o. It is worth noting that RANS turbulencemodels that use a linear Boussinesq closure expression are unable to produce Reynolds stresses that are not alignedwith the mean strain rate. Several authors (see, for example, Churchfield and Blaisdell [10]) have proposed thatthis weakness in the linear Boussinesq approximation makes a key contribution to RANS models diffusing tipvortices much more quickly than is seen in experiments. The present simulation shows the expected 45o offsetbetween the lobes of S23 and τ23 once the vortex rollup is complete approximately one chord downstream of thetrailing edge.

Future Work

The remaining terms in the TKE equation are to be extracted from the flow solution and explored in detail. Thiswill include determining the dominant terms in the turbulent energy budget as the vortex evolves downstream ofthe trailing edge.

Furthermore, the pressure variance, (p′)2rms will be compared with a common approximate expression for thepressure fluctuations, p′ ∼ 0.39ρk. This approximation is appropriate for isotropic, homogeneous turbulence;however, it will be seen that it does not adequately capture the character of the turbulent pressure fluctuationswithin the highly ordered, anisotropic vortex core. This has implications for RANS-based cavitation models.

References

[1] Barnes W McCormick, Jr. A Study of the Minimum Pressure in a Trailing Vortex System. Doctoral disserta-tion, Pennsylvania State University, 1954.

[2] K Sinding and M Krane. Tip vortex core pressure estimates derived from velocity field measurements. InAPS Division of Fluid Dynamics Meeting Abstracts, 2016.

[3] T Brockett. Minimum pressure envelopes for modified NACA-66 sections with NACA A= 0.8 camber andbuships type 1 and type 2 sections. Technical report, David Taylor Model Basin, 1966.

[4] David Hanson. Cavitation Inception for Non-Elliptically Loaded Fins. Master’s thesis, The PennsylvaniaState University, 2012.

[5] Siemens. Star CCM+ v 12.04 User Manual, 2017.

6

[6] F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient tensor.Flow, Turbulence and Combustion, 62(3):183–200, 1999.

[7] Nicholas J Georgiadis, Donald P Rizzetta, and Christer Fureby. Large-Eddy Simulation: Current Capabili-ties, Recommended Practices, and Future Research. AIAA Journal, 48(8):1772–1784, 2010.

[8] Johan Larsson, Soshi Kawai, Julien Bodart, and Ivan Bermejo-Moreno. Large eddy simulation with modeledwall-stress: recent progress and future directions. Bulletin of the JSME Mechanical Engineering Reviews,3(1), 2016.

[9] Jim Chow, Greg Zilliac, and Peter Bradshaw. Turbulence Measurements in the Near Field of a WingtipVortex. 1997.

[10] Matthew J Churchfield and Gregory A Blaisdell. Numerical Simulations of a Wingtip Vortex in the NearField. Journal of Aircraft, 46(1):230–243, 2009.

7